Open Access
The Fractional Poisson Process and the Inverse Stable Subordinator
Ear,Nih grant R Eb +1 more
TLDR
In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.Abstract:
The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.read more
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Time-inhomogeneous jump processes and variable order operators
TL;DR: In this article, the authors introduce non-decreasing jump processes with independent and time non-homogeneous increments, which generalize subordinators in the sense that their Laplace exponents are possibly different Bernstein functions for each time $t.
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Randomly Stopped Nonlinear Fractional Birth Processes
Enzo Orsingher,Federico Polito +1 more
TL;DR: In this paper, the nonlinear classical pure birth process and the fractional pure birth processes subordinated to various random times were analyzed and the state probability distribution was derived, and the corresponding governing differential equation was presented.
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Fractional Poisson fields
Nikolai Leonenko,Ely Merzbach +1 more
TL;DR: Using inverse subordinators and Mittag-Leffler functions, this paper presented a new definition of a fractional Poisson process parametrized by points of the Euclidean space.
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Space-fractional versions of the negative binomial and Polya-type processes
TL;DR: In this article, a space fractional negative binomial process (SFNB) was introduced by time-changing the Space fractional Poisson process by a gamma subordinator and its one-dimensional distributions were derived in terms of generalized Wright functions and their governing equations were obtained.
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Tempered Mittag-Leffler Lévy processes
TL;DR: In this article, a tempered Mittag-Leffler Levy process (TMLLP) is represented as a tempered stable subordinator delayed by a gamma process and its probability density is defined.
References
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Journal ArticleDOI
Correlation Structure of Time-Changed Lévy Processes
TL;DR: In this article, the correlation function for time-changed L evy processes has been studied in the context of continuous time random walks, where the second-order correlation function of a continuous-time random walk is defined.
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Applications of inverse tempered stable subordinators
TL;DR: This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its “folded” density solves a temperamental telegraph equation.
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Time-changed Poisson processes
TL;DR: In this article, the authors considered time-changed Poisson processes and derived the governing difference-differential equations (DDEs) for these processes, and derived a new governing partial differential equation for the tempered stable subordinator of index 0 β 1.
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Fractional Skellam processes with applications to finance
TL;DR: In this paper, the authors define fractional Skellam processes via the time changes in Poisson and Skekam processes by an inverse of a standard stable subordinator.
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Inverse Tempered Stable Subordinators
TL;DR: In this paper, the first-hitting time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator is considered, and the limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained.